Integrable Dynamics of Charges Related to the Bilinear Hypergeometric Equation

A family of systems related to a linear and bilinear evolution of roots of polynomials in the complex plane is introduced. Restricted to the line, the evolution induces dynamics of the Coulomb charges (or point vortices) in external potentials, while its fixed points correspond to equilibriums of charges in the plane. The construction reveals a direct connection with the theories of the Calogero-Moser systems and Lie-algebraic differential operators. A study of the equilibrium configurations amounts in a construction (bilinear hypergeometric equation) for which the classical orthogonal and the Adler-Moser polynomials represent some particular cases.